Sterling's 6.6 per cent advance this year is vexing the Bank, whose monetary policy is aimed at keeping inflation at 2.5 per cent.
As a strong pound causes import prices to fall and export orders to dwindle, inflation is starting to undershoot the target.
To keep prices close to the target, the Bank's Monetary Policy Committee (MPC) may cut rates, which would have the dual effect of stimulating the economy and weakening the pound.
It has already cut rates at six of its past eight meetings.
"Sterling could appreciate further," said Derek Halpenny, a currency economist at Bank of Tokyo-Mitsubishi. "That's going to allow the MPC to cut."
He predicts the euro will fall to 64.5p by August, leading the Bank to lower its securities repurchase rate a quarter point to 5 per cent.
On Friday, the pound fell as low as $1.6004, its weakest since April 8, and down more than a cent from a week ago. It was little changed at pounds 0.6595 per euro.
Reports last week showed cheaper borrowing is helping the economy.
The jobless rate came in at a 19-year low, and annual wage growth increased by an average 4.8 per cent. Britain's broad money supply rose 1.2 per cent in April - the largest increase since July last year.
A drop in retail sales, though, showed that, even after rate reductions totalling 225 basis points since October, signs of weakness persist.
"The economy shows signs of picking up, but it's still not barrelling along," said Sean Callow, a currency analyst at IDEAglobal.com.
The pound has not been untouched since the rate cuts began - it has fallen almost 7 per cent against the dollar since 8 October, when the Bank of England took its first bite out of the benchmark rate.
The cuts since then have whittled away at the return on pound deposits and closed the gap with the US, where the key rate stands at 4.75 per cent.
Still, the slide in borrowing costs has not stopped the pound from gaining against the euro. At 5.25 per cent, rates are more than double the 2.5 per cent benchmark rate in the euro zone.
A faltering economy, combined with a two-month old war in Kosovo, is making investors skittish about euro-denominated assets. A rate cut in the UK will not undermine the pound until the euro region shows signs of recovery.
"It's down to Europe," said Tokyo-Mitsubishi's Halpenny. He added that recent figures, such as waning German business confidence, are "pointing to further weakness - that's going to underpin sterling".
The Bank of England Governor, Eddie George, will get a chance to air his concerns about the pound's drag on inflation when he testifies on Tuesday before the House of Commons Treasury Select Committee. He'll discuss the Bank's quarterly inflation report.
In that report, the Bank said a strong pound may lead to lower rates, the same warning it gave on 6 May - after announcing it, the Bank left rates untouched for only the second month since October.